forked from apvijay/convex_opt
-
Notifications
You must be signed in to change notification settings - Fork 0
/
misc.tex
47 lines (45 loc) · 1.1 KB
/
misc.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
\noindent
\textit{Misc}\\[5mm]
\begin{align*}
\log det X &= tr(logX)\\
- \log det X^{-1} &= \log det X\\
\mbox{Schur complement } S &= C - B^TA^{-1}B \quad \mbox{for }
X = \begin{bmatrix} A&B\\B^T&C \end{bmatrix} \\
X \succ 0 \mbox{ iff } &A \succ 0 \mbox{ and } S \succ 0\\
A \succ 0 \implies &X \succeq 0 \mbox{ iff } S \succeq 0
\end{align*}
%
%
\begin{table}[ht]
\begin{tabular}{|>{$}l<{$}|>{$}l<{$}|}
\hline
\multicolumn{2}{|>{$}c<{$}|}{
\partial / \partial X operations
}\\ \hline
& \\
a^TXb & ab^T\\
a^TX^Tb & ba^T\\
b^TX^TXc & X(bc^T+cb^T)\\
det(X) & det(X) (X^{-1})^T\\
det(AXB) & det(AXB) (X^{-1})^T\\
tr(X) & I\\
tr(XA) & A^T\\
tr(AXB) & A^TB^T\\
tr(AX^TB) & BA\\
tr(X^TA) & A\\
tr(A^TX) & A\\
tr(X^2) & 2X^T\\
tr(X^TX) & 2X\\
\ln det(X) & X^{-1}\\
\prod eig(X) & det(X) (X^{-1})^T\\
\sum eig(X) & I\\
tr(X^{-1}A) & -X^{-1} A^T X^{-1}\\ \hline
\end{tabular}
\end{table}
% \rule{\linewidth}{0.1mm}
\begin{align*}
\|z\|_a &= \sup\{u^T z | \|u\|_{a*} = 1\}\\
\|X\|_{a,b} &= \sup_{v \ne 0} \frac{\|Xv\|_a}{\|v\|_b}\\
&=\sup \{\|Xv\|_a | \|v\|_b = 1\}\\
&=\sup \{ u^TXv | \|u\|_{a*} = 1, \|v\|_b = 1\}
\end{align*}